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A supermassive scalar star at the Galactic Center?

Diego F. TorresInstituto Argentino de Radioastronomıa, C.C.5, 1894 Villa Elisa, Buenos Aires, Argentina

dtorres@venus.fisica.unlp.edu.ar

and

S. Capozziello and G. LambiaseDipartimento di Scienze Fisiche E.R. Caianiello, Universita di Salerno, 84081 Baronissi (SA), Italy

Istituto Nazionale di Fisica Nucleare, Sez. Napoli, Italy

capozziello@sa.infn.it

lambiase@sa.infn.it

ABSTRACT

We explore whether supermassive non-baryonic stars (in particular boson, mini-boson andnon-topological soliton stars) might be at the center of some galaxies, with special attention tothe Milky Way. We analyze, from a dynamical point of view, what current observational datashow, concluding that they are compatible with a single supermassive object without requiringit to be a black hole. Particularly, we show that scalar stars fit very well into these dynamicalrequirements. The parameters of different models of scalar stars necessary to reproduce theinferred central mass are derived, and the possible existence of boson particles with the adequaterange of masses is commented. Accretion to boson stars is also analyzed, and a comparison withanother non-baryonic candidate, a massive neutrino ball, which is also claimed as an alternativeto the central black hole, is given. Both models are capable to explain the nature of the objectin Sgr A∗ without invoking the presence of a singularity. One difficult issue is why the accretedmaterials will not finally produce, in a sufficiently long time, a black hole. We provide ananswer based on stellar disruption in the case of boson stars, and comment several suggestionsfor its possible solution in neutrino ball scenarios. Finally, we discuss the prospects for theobservational detection of these supermassive scalar objects, using the new generation of X-rayand radio interferometry satellites.

Subject headings: galaxies: nuclei — stars: boson stars — black holes

1. Introduction

During the last years, the possible existence of asingle large mass in the Galactic Center has beenfavored as the upper bound on its size tightens,and stability criteria rule out complex clusters. Al-though it is commonly believed that this centralmass is a supermassive black hole, it is not yetestablished, as we discuss below, on a firm obser-vational basis.

The aim of this paper is to present an alterna-

tive model for the supermassive dark object in thecenter of our Galaxy, formed by self-gravitatingnon-baryonic matter composed by bosons. Thiskind of objects, so-called boson stars, are wellknown to physicists, but up to now, observationalastrophysical consequences have hardly been ex-plored. The main characteristics of this model are:

1. It is highly relativistic, with a size com-parable to (but slightly larger than) theSchwarzschild radius of a black hole of equalmass.

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2. It has neither an event horizon nor a sin-gularity, and after a physical radius isreached, the mass distribution exponentiallydecreases.

3. The particles that form the object interactbetween each other only gravitationally, insuch a way that there is no solid surface towhich falling particles can collide.

It is the purpose of this work to show thatthese features are able to produce a Galactic Cen-ter model which can be confronted with all knownobservational constraints, and to point the way inwhich such a center could be differentiated from ausual supermassive black hole.

1.1. Why scalar fields?

Interesting models for dark matter use weaklyinteracting bosons, and primordial nucleosynthesisshow that most of the mass in the universe shouldbe non-baryonic if Ω ∼ 1. Most models of inflationmake use of scalar fields. Scalar-tensor gravitationis the most interesting alternative to general rel-ativity. Recent results from supernovae, which inprinciple were thought of to favor a cosmologicalconstant (Perlmutter et al. 1999), can as well besupported by a variety of models, some of themwith scalar fields too (Celerier 1999). Particlephysicist expect to detect the scalar Higgs particlein the next generation of accelerators. Scalar dila-tons appear in low energy unified theories, wherethe tensor field gµν of gravity is accompanied byone or several scalar fields, and in string effectivesuper-gravity. The axion is a scalar with a longhistory as a dark matter candidate, and Goldstonebosons have also already inferred masses. Sym-metry arguments, which once led to the conceptof neutron stars, may force to ask whether therecould be stellar structures made up of bosons in-stead of fermions.

In recent works, Schunck and Liddle (1997),Schunck and Torres (2000), and Capozziello, Lam-biase and Torres (2000) analyzed some of the ob-servational properties of boson stars, and foundthem notoriously similar to black holes. In par-ticular, the gravitational redshift of the radiationemitted within a boson star potential, and the ro-tational curves of accreted particles, were studiedto assess possible boson star detection. The in-terest in observational properties of boson stars

also led to investigate them as a possible lens ina gravitational lensing configuration (Dabrowskiand Schunck 2000). Recent studies are puttingforward the gravitational lensing phenomenon instrong field regimes (Virbhadra et al. 1998; Virb-hadra and Ellis 2000; Torres et al. 1998a,b). Thiswould be the case for boson stars, which are gen-uine relativistic objects.

This work is organized as follows. Sec. II is abrief summary of the main observational resultsconcerning Sgr A∗, and the main hypotheses inorder to explain it are outlined there. Sec. III an-alyzes what dynamical observational data show,and what kind of models can support them. Sec.IV gives the basic ingredients to theoretically con-struct scalar stars, shows mass and radius esti-mates, and study effective potentials and orbitsof particles. In Sec. V we study the center of theGalaxy, show which are the stable scalar star mod-els able to fit such a huge mass, and comment onthe possible existence of boson particles with therequired features. We also provide there an as-sessment of the disruption processes in boson starscenarios. Discussion and conclusions are given inSections VI and VII.

2. The Galactic Center

2.1. Main observational facts

The Galactic Center is a very active region to-ward the Sagittarius constellation where, at least,six very energetic radio sources are present (SgrA, A∗, B1, B2, C, and D). Furthermore, there areseveral supernova remnants, filaments, and veryreach star clusters. Several observational cam-paigns (Genzel et al. 1994) have identified the ex-act center with the supermassive compact darkobject in Sgr A∗, an extremely loud radio source.The mass and the size of the object has been estab-lished to be (2.61 ± 0.76) × 106M⊙ concentratedwithin a radius of 0.016 pc (about 30 lds)(Ghez1998; Genzel et al. 1996).

More precisely, Ghez et al. (1998) have madeextremely accurate velocity measurements in thecentral square arcsecs. From this bulk of data, itis possible to state that a supermassive compactdark object is present at the Galactic Center. Itis revealed by the motion of stars moving withinprojected distances around 0.01 pc from the radiosource Sgr A∗, at projected velocities in excess of

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1000 km s−1. In other words, a high increase inthe velocity dispersion of the stars toward the dy-namical center is revealed. Furthermore, a largeand coherent counter–rotation, especially of theearly–type stars, is shown, supporting their originin a well–defined epoch of star formation. Obser-vations of stellar winds nearby Sgr A∗ give a massaccretion rate of dM/dt = 6×10−6M⊙ yr−1 (Gen-zel et al. 1996). Hence, the dark mass must havea density ∼ 109M⊙ pc−3 or greater, and a mass–to–luminosity ratio of at least 100M⊙/L⊙. Thebottom line is that the central dark mass seems tobe a single object, and that it is statistically verysignificant (∼ 6 − 8σ).

2.2. Domain of the black hole?

Such a large density contrast excludes that thedark mass could be a cluster of almost 2 × 106

neutron stars or white dwarfs. Detailed calcu-lations of evaporation and collision mechanismsgive maximal lifetimes of the order of 108 years,much shorter than the estimated age of the Galaxy(Sanders 1992; Maoz 1995).

As a first conclusion, several authors state thatin the Galactic Center there is either a single su-permassive black hole or a very compact clusterof stellar size black holes (Genzel et al. 1996). Weshall come back to this paradigm through the restof the paper (particularly in Section III).

Dynamical evidence for central dark objects hasbeen published for 17 galaxies, like M87 (Fordet al. 1994; Macchetto et al. 1997), or NGC4258(Greenhill et al. 1995), but proofs that they re-ally are black holes requires measurement of rel-ativistic velocities near the Schwarzschild radius,rs ≃ 2M•/(108 M⊙) AU (Kormendy and Rich-stone 1995), or seeing the properties of the ac-cretion disk. Due to the above mentioned massaccretion rate, if Sgr A∗ is a supermassive blackhole, its luminosity should be more than 1040ergs−1, provided the radiative efficiency is about 10%.On the contrary, observations give a bolometricluminosity less than 1037 erg s−1, already takinginto account the luminosity extinction due to in-terstellar gas and dust. This discrepancy is theso–called “blackness problem” which has led to thenotion of a “black hole on starvation”. Standarddynamics and thermodynamics of the spherical ac-cretion onto a black hole must be modified in or-der to obtain successful models (Falcke and Hein-

rich 1994). Recent observations, we recall, probethe gravitational potential at a radius larger than4×104 Schwarzschild radii of a black hole of mass2.6 × 106M⊙ (Ghez 1998).

2.3. Domain of the massive neutrino?

An alternative model for the supermassive com-pact object in the center of our Galaxy has beenrecently proposed by Tsiklauri and Viollier (1998).The main ingredient of the proposal is that thedark matter at the center of the galaxy is non-baryonic (but in any case fermions, e.g. massiveneutrinos or gravitinos), interacting gravitation-ally to form supermassive balls in which the de-generacy pressure balances their self–gravity. Suchneutrino balls could have formed in early epochs,during a first–order gravitational phase transitionand their dynamics could be reconciled, with someadjustments, to the Standard Model of Cosmology.

Several experiments are today running tosearch for neutrino oscillations. LSND (White1998) found evidence for oscillations in the νe−νµchannel for pion decay at rest and in flight. Onthe contrary KARMEN (Zeitnitz 1998) seems tobe in contradiction with LSND evidence. CHO-RUS and NOMAD at CERN are just finishing thephase of 1994–1995 data analysis. It is very likelythat exact predictions for and νµ − ντ oscillationswill be available in next years. Thanks to the factthat it is possible to give correct values for themasses to the quarks up, charm, and top, it ispossible to infer reasonable values of mass for νe,νµ, and ντ . In order to explain the characteristicsof Sgr A∗, we need fermions whose masses rangebetween 10 and 25 keV which, cosmologically, fallinto the category of warm dark matter. It is in-teresting to note that a good estimated value forthe mass of τ -neutrino is

mντ= mνµ

(

mt

mc

)2

≃ 14.4 keV . (1)

Choosing fermions like neutrinos or gravitinos inthis mass range allows for the formation of su-permassive degenerate objects from 106M⊙ to109M⊙.

The theory of heavy neutrino condensates,bound by gravity, can be easily sketched consider-ing a Thomas–Fermi model for fermions (Viollieret al. 1992). We can set the Fermi energy equal

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to the gravitational potential which binds the sys-tem, and see that the number density is a functionof the gravitational potential. Such a gravitationalpotential will obey a Poisson equation where neu-trinos (and anti-neutrinos) are the source term.This equation is valid everywhere except at theorigin. By a little algebra, the Poisson equationreduces itself to the radial Lane–Emden differ-ential equation with polytropic index n = 3/2,which is equivalent to the Thomas–Fermi differen-tial equation of atomic physics, except for a minussign that is due to the gravitational attraction ofthe neutrinos. The Newtonian potential, which isthe solution of such an equation, is Φ(r) ∼ r−4.Considering a standard accretion disk, if Sgr A∗

is a neutrino star with radius R = 30.3 lds (∼ 105

Schwarzschild radii), mass MGC = 2.6 × 106M⊙,and luminosity LGC ∼ 1037erg sec−1, it shouldconsist of neutrinos with masses m ≥ 12.0 keVfor gν = 4, or m ≥ 14.3 keV for gν = 2 (Bilic1998; Viollier et al. 1992). It is specially ap-pealing that with the same mass for neutrinos,several galactic centers can be modeled. For in-stance, similar results hold also for the dark object(M ∼ 3 × 109M⊙) inside the center of M87. Dueto the Thomas–Fermi theory, the model fails atthe origin: we have to consider the effect of thesurrounding baryonic matter which, in some sense,has to stabilize the neutrino condensate. In fact,the solution Φ(r) ∼ r−4 is clearly unbounded frombelow.

The model proceeds assuming a thermodynam-ical phase where a constant neutrino number den-sity can be taken into consideration. This is quitenatural for a Fermi gas at temperature T = 0.The Poisson equation can be recast in a Lane–Emden form with polytropic index n = 0. Thesolution of such an equation is Φ(r) ∼ r2, whichis clearly bounded (Capozziello and Iovane 1999).For T 6= 0, we get a solution of the form Φ(r) ∼r−4. Matching these two results it is possible toconfine the neutrino ball. On the other hand, asimilar result is recovered using the Newton The-orem for a spherically symmetric distribution ofmatter of radius R (Binney and Tremaine 1987).In that case, the potential goes quadratic insidethe sphere while it goes as Φ(r) ∼ r−1 match-ing on the boundary. In our case, the situationis similar assuming the matching with a steeperpotential.

If such a neutrino condensate exists in the cen-ter of Galaxy, it could act as a spherical thick lens(a magnifying glass) for the stars behind it, so thattheir apparent velocities will be larger than in real-ity. In other words, depending on the line of sight,it should be possible to correct the projected ve-locities by a gravitational lensing contribution, sotrying to explain the bimodal distribution (earlyand late type stars) actually observed (Ghez 1998;Genzel et al. 1996). Since the astrophysical fea-tures of the object in Sgr A∗ are quite well known,accurate observations by lensing could contributeto the exact determination of particle constituents.A detailed model and comparison with the datawas presented by some of us (Capozziello and Io-vane 1999).

3. Observational status

3.1. Brief review on dynamical data and

what are they implying

An important information on the central ob-jects of galaxies (particularly in active galactic nu-clei) is the short timescale of variability. This hasthe significance of putting an upper bound –knownas causality constraint– on the size of the emittingregion: If a system of size L suddenly increases itsemissivity at all points, the temporal width withwhich we receive it is L/c and thus, a source cannot fluctuate in a way that involves its entire vol-ume in timescales shorter than this (unless c is nota limiting velocity). This finally yields a corre-sponding maximum lenghscale, typically between10−4 to 10 pc (Krolik 1999). Autocorrelation inthe emission argue against the existence of a clus-ter of objects (unless only one member dominatesthe emissivity), and the small region in which thecluster should be allows two body encounters tobe very common and produces the lost of the sta-bility of complex clusters. These facts can be usedto conclude that a single massive object must bein the center of most galaxies. What observationsshow in this case is that the size of the emittingregions are very small. This does not directly im-plicate black holes as such.

The way in which we expect to detect the influ-ence of a very massive object is through its gravity.The “sphere of influence” of a large mass is definedby the distance at which its potential significantlyaffects the orbital motions of stars and gas, and is

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given by

R∗ = GM/σ2∗ ∼ 4M7σ

−2∗,100pc, (2)

where the central mass is normalized to 107M⊙

and σ∗,100 is a rms orbital speed in units of 100km s−1. Thus, even very large masses have a smallsphere of influence. Within this sphere, the ex-pected response to a large nuclear mass can bedivided in two groups. Firstly, the response of in-terstellar gas can hardly be anything different froman isotropic motion in the center of mass frame.Random speeds are often thought to be muchsmaller than this overall speed, which far from thesphere of influence is just given by (GM/r)1/2. Ifthe energy produced by random motions is radi-ated away, the gas behaves as a whole and tend toflatten itself, conserving the angular momentum.Observations of the rotational velocity of gas as afunction of the radius can thus provide a measureof the total mass at the center. Then, what ob-servation searches is the existence of a Keplerianpotential signature in the flattened gas velocity dis-tribution. Examples of this are the radio galaxiesM87 (Ford et al. 1994), NGC 4258 (Miyoshi et al.1995), and many others. Then, any massive objectproducing a velocity distribution with a Kepleriandecrease would be allowed by observation.

Secondly, we consider the response of the stars.In this case, peculiar velocities are larger or com-parable to the bulk streaming and it is harder toactually differentiate the stellar response to a nu-clear mass from the observable properties of a purestellar potential. However, stellar motions, con-trary to that of gas, is not affected by other forces(as those produced by magnetic fields), and arebetter tracers of mass distributions. What obser-vations show in this case are the stellar densityand the velocity distribution. Use of the collision-less Boltzmann equation allows one to get Jean’srelation (Binney and Tremaine 1987; Krolik 1999):

GM(r)

r= V 2

rot−σ2r

[

d lnn(r)

d ln r− d lnσ2

r

d ln r+ 2 − σ2

t

σ2r

]

(3)here, n(r) is the spherically symmetric distribu-tion of stars, Vrot is the mean rotational speed,σr,θ,φ are velocity dispersions and σ2

t = σ2θ + σ2

φ.Measurements of V and random velocities deter-mine that M(r) decline inwards until a critical ra-dius, where M becomes constant. What really

happens, however, if that even if this region is notscrutinized, the mass to light ratio becomes suf-ficiently large to suggest the presence of a largedark mass. A combination of this with other mea-surements strongly suggest that this dark mass isa single object.

To summarize, usual measurements point toinform us that there exist a single supermassiveobject in the center of some galaxies but do notdefinitively state its nature. As Kormendy andRichstone (1995) have stated, the black hole sce-nario has become our paradigm. But suggestionsthat the dark objects are black holes are basedonly on indirect astrophysical arguments, and sur-prises are possible on the way to the center.

3.2. The Galaxy

We follow Genzel et al. (1996) and parameter-ize the stellar density distribution as,

n(r) =Σ0

R0

1

1 + (R/R0)α. (4)

Note that R0 is related to the core radius throughRcore = b(α)R0. Genzel et al. found that thebest fit parameters for the observed stellar clusterare a central density of 4 × 106M⊙pc−3, a coreradius of 0.38 pc, and a value α = 1.8 (b(α) =2.19). With this distribution, they found that adark mass of about 2.5 ×106M⊙ was needed to fitthe observational data.

The cluster distribution, and the cluster plusa black hole constant mass is shown in Fig. 1.Black boxes represent Genzel et al. 1996 (Table10) and Eckart and Genzel 1997 (Fig. 5) data. Adashed line stands for the stellar cluster contribu-tion, while a dot-dashed line represent an enclosedpoint-like black hole mass. The solid line in theright half of the figure stands for the mass dis-tribution both for a black hole and a boson starplus the stellar cluster. The mass dependence weare plotting for the boson star is obtained in Sec-tion IV, and represents the mass distribution of amini-boson star (Λ = 0, m[GeV] = 2.81 × 10−26)with dimensionless central density σ(0) equal to0.1. Other boson star configurations, with appro-priate choice of the boson field massm, yield to thesame results. Note the break of more than threeorders of magnitude in the x-axis, this is causedbecause the boson star distribution, further out ofthe equivalent Schwarzschild radius, behaves as a

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black hole. It begins to differ from the black holecase at radius more than three orders of magnitudeless than the innermost data point, that is why thex-axis has to have a break. From the mass distri-bution, a boson star in the center of the galaxy isvirtually indistinguishable from a black hole.

Tsiklauri and Viollier (1998) have shown thatthe same observational data can also be fitted us-ing an extended neutrino ball. In that case, dif-ferences begin to be noticed just around the in-nermost data point. It is then hard to determinewhether the central object is a black hole, a neu-trino ball, or a boson star based only on dynamicaldata now at hand, and other problems concerningthe accretion disk have to considered (see below).Even harder is the situation for deciding –usingonly this kind of data– if the supermassive objectis a boson star instead of a black hole: as a bosonstar is a relativistic object, the decay of the en-closed mass curve happens close the center. This,however, will provide an equivalent picture than ablack hole for disruption and accretion processes;we shall comment on it in the next sections.

To apply Eq. (3) to the observational datawe have to convert intrinsic velocity dispersions(σr(r)) and volume densities (n(r)) to projectedones (Binney and Tremaine 1987), these are theones we observe. We shall also consider thatδ = 1− σ2

θ/σ2r , the anisotropy parameter, is equal

to 0, and we are assuming implicitly that σθ = σφ.We take into account the following Abel integrals,

Σ(p) = 2

∫ ∞

p

n(r)r√

r2 − p2dr, (5)

Σ(p)σr(p) = 2

∫ ∞

p

n(r)σr(r)2r

√

r2 − p2dr. (6)

Σ(p) denotes surface density, σr(p) is the pro-jected velocity dispersion, and p is the projecteddistance. We adopt Genzel et al.’s (1996) andTsiklauri and Viollier’s (1998) parameterizationfor σr(r),

σr(r) = σ(∞)2 + σ(2′′)2(

R

2′′

)−2β

. (7)

What one usually does is to numerically inte-grate Eqs. (5,6) and fit the observational data(σr(p) vs. p). To do so one also has to assume adensity dependence for the cluster (as in Eq. (4)),

and the dark mass. With a point-like dark massof 2.5 ×106M⊙, the parameters of the fit result tobe σ(∞) = 55 km s−1, σ(2′′) = 350 km s−1, andβ = 0.95.

If one now changes the central black hole fora neutrino ball, or a boson star, one has to con-sider the particular density dependence for theseobjects. In this way, n(r) = nstellar cluster(r) +ndark mass(r). We obtained nboson star(r) asM(r)/(4/3πr3),where M(r) is the fitted mass dependence of theboson star as explained in the next section. Itis now useful to consider that the fitting of σr(p)is made taking into account observational datapoints in regions where the boson star generatedspace-time is practically indistinguishable from ablack hole. Then, we may expect that the actualparameters, σ(∞), σ(2′′), and β, will be very closeto those obtained for the black hole. For our pur-poses, it is enough to take the same σr(r) as inthe black hole case, and compute σr boson star(p)using Eqs. (5,6), with the adequate total n(r).

In Fig. 2 we show the observational dataof Eckart and Genzel (1997) (filled black boxes)and Genzel et al. (1996) (hollow circles) super-imposed with the curve for σr(p) that we ob-tain with a mini-boson star (Λ = 0, σ(0) = 0.1,m[GeV] = 2.81 × 10−26) in the galactic center.Other boson star configurations, with appropri-ate choice of the boson field mass m, yield tothe same results. For this configuration, we used,as will be explained in Section IV, a mass distri-bution given by a Boltzmann-like equation withA2 = 2.5 × 106M⊙, A1 = −0.237 × 106M⊙,R0 = 1.19210−6 pc, and ∆R = 4.16310−6 pc. Inthe range plotted, and in which data is available,the differences between boson and black hole theo-retical curves is undetectable. They only begin todeviate from each other at p ∼ 10−4 pc, well be-yond the last observational data point. Even thedeviation in such a region is as slight as 1 km s−1,and it only becomes more pronounced when p val-ues are closer to the center. However, we shouldrecall that it has no sense to go to such extremevalues of p: the stars will be disrupted by tidalforces (see the discussion in Section VI) in thoseregions.

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4. Boson stars

4.1. Basic concepts and configurations

Let us study the Lagrangian density of a mas-sive complex self-gravitating scalar field (taking~ = c = 1),

L =1

2

√

| g |[

m2Pl

8πR+ ∂µψ

∗∂µψ − U(|ψ|2)]

,

(8)where R is the scalar of curvature, |g| the modulusof the determinant of the metric gµν , and ψ is acomplex scalar field with potential U . Using thisLagrangian as the matter sector of the theory, weget the standard field equations,

Rµν −1

2gµνR = − 8π

m2Pl

Tµν(ψ) , (9)

2ψ +dU

d|ψ|2ψ = 0 , (10)

where the stress energy tensor is given by,

Tµν = (∂µψ∗)(∂νψ) +

−1

2gµν

[

gαβ(∂αψ∗)(∂βψ) − U(|ψ|2)

]

(11)

and2 = ∂µ

[

√

| g | gµν∂ν]

/√

| g | (12)

is the covariant d’Alembertian. Because of thefact that the potential is a function of the squareof the modulus of the field, we obtain a global U(1)symmetry. This symmetry, as we shall later dis-cuss, is related with the conserved number of par-ticles. The particular form of the potential is whatmakes the difference between mini-boson, boson,and soliton stars. Conventionally, when the po-tential is given by

U = m2|ψ|2 +λ

2|ψ|4 , (13)

where m is the scalar mass and λ a dimensionlessconstant measuring the self-interaction strength,mini-boson stars are those spherically symmetricequilibrium configurations with λ = 0. Bosonstars, on the contrary, have a non-null value of λ.The previous potential with λ 6= 0 was introducedby Colpi et al. (1986), who numerically found thatthe masses and radius of the configurations were

deeply enlarged in comparison to the mini-bosoncase.

Soliton (also called non-topological soliton)stars are different in the sense that, apart fromthe requirement that the Lagrangian must be in-variant under a global U(1) transformation, itis required that –in the absence of gravity– thetheory must have non-topological solutions; i.e.solutions with a finite mass, confined to a finiteregion of space, and non-dispersive. An exampleof this kind of potentials is the one introduced byLee and his coworkers (Friedberg et al. 1987),

U = m2|ψ|2(

1 − |ψ|2Φ2

0

)2

, (14)

where Φ0 is a constant. In general, boson starsaccomplish the requirement of invariance under aU(1) global transformation but not the solitonicsecond requirement. To fulfill it, it is necessarythat the potential contains attractive terms. Thisis why the coefficient of (ψ∗ψ)2 of the Lee form hasa negative sign. Finally, when |ψ| → ∞, U mustbe positive, which leads, minimally, to a sixth or-der function of ψ for the self-interaction. It is usu-ally assumed, because of the range of masses andradius for soliton stars in equilibrium, that theyare huge and heavy objects, although this finallydepends on the choice of the different parameters.

We shall now briefly explain how these con-figurations can be obtained (Kaup 1968; Ruffiniand Bonazzola 1969; Lee and Pang 1992; Liddleand Madsen 1992; Mielke and Schunck 1998). Weadopt a spherically symmetric line element

ds2 = eν(r)dt2 − eµ(r)dr2 − r2(dϑ2 + sin2 ϑ dϕ2) ,(15)

with a scalar field time-dependence ansatz consis-tent with this metric:

ψ(r, t) = σ(r)e−iωt (16)

where ω is the (eigen-)frequency. This form of thefield ensures us to be working in the configurationsof minimal energy (Friedberg et al. 1987).

The non-vanishing components of the energy-momentum tensor are

T00 = ρ =

1

2[ω2σ2(r)e−ν + σ′2(r)e−µ + U ] , (17)

T11 = pr =

1

2[ω2σ2(r)e−ν + σ′2(r)e−µ − U ] , (18)

T22 = T3

3 = p⊥ = −1

2[ω2σ2(r)e−ν − σ′2(r)e−µ − U ] (19)

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where ′ = d/dr. One interesting characteristicof this system is that the pressure is anisotropic;thus, there are two equations of state pr = ρ− Uand p⊥ = ρ − U − σ′2(r)e−µ. The non-vanishingindependent components of the Einstein equationare

ν′ + µ′ =8π

m2Pl

(ρ+ pr)reµ , (20)

µ′ =8π

m2Pl

ρreµ − 1

r(eµ − 1) . (21)

Finally, the scalar field equation is

σ′′ +

(

ν′ − µ′

2+

2

r

)

σ′ + eµ−νω2σ− eµdU

dσ2σ = 0 .

(22)To do numerical computations and order of mag-nitude estimates, it is useful to have a new set ofdimensionless variables. We adopt here

x = mr, (23)

for the radial distance, we redefine the radial partof the boson field as

σ =√

4π σ/mPl, (24)

and introduce

Λ = λm2Pl/4πm

2, Ω =ω

m. (25)

In order to obtain solutions which are regular atthe origin, we must impose the following boundaryconditions σ′(0) = 0 and µ(0) = 0. These solu-tions have two fundamental parameters: the self-interaction and the central density (represented bythe value of the scalar field at the center of thestar). The mass of the scalar field fixes the scaleof the problem. Boundary conditions representingasymptotic flatness must be applied upon the met-ric potentials, these determine –what is actuallyaccomplished via a numerical shooting method–the initial value of ν = ν(0). Then, having de-fined the value of the self interaction, or alterna-tively, the form of the soliton potential, the equi-librium configurations are parameterized by thecentral value of the boson field. As this centralvalue increases, so does the mass and radius of thethe star. This happens until a maximum valueis reached in which the star looses its stabilityand disperses away (the binding energy being pos-itive). Up to this value of σ0, catastrophe theory

can be used to show that these equilibrium config-urations are stable (Kusmartsev et al. 1991). Asan example of boson star configurations, we showin Fig. 3, the mass and number of particles (seebelow) for a Λ = 10 Colpi et al.’s potential, andin Fig. 4, the stability analysis.

When Λ ≫ 1, as in some of the cases we explorein the next sections, we must follow an alternativeadimensionalization (Colpi et al. 1986). For largeΛ, we shift to the following set of variables,

σ∗ = σΛ1/2, (26)

x∗ = xΛ−1/2, (27)

M∗ = MΛ−1/2. (28)

Here, M∗ is defined by

eµ =

(

1 − 2M∗

x∗

)−1

, (29)

which corresponds to the Schwarzschild mass (seebelow). Ignoring terms O(Λ−1), the scalar waveequation is solved algebraically to yield,

σ∗ = (Ω2e−ν − 1)1/2, (30)

and up to the same accuracy, the field equationsare

dM∗

dx∗=

1

4x2∗(3Ω2e−ν + 1)(Ω2e−ν − 1), (31)

dν

dx∗

e−µ

x∗− 1

x

2

∗

(1 − e−µ) =1

2(Ω2e−ν − 1)2. (32)

The system now depends only on one free param-eter Ω2e−ν(0). Numerical solutions show that themaximum mass corresponding to a stable star isgiven by Mmax ∼ 0.22Λ1/2m2

Pl/m.

4.2. Masses estimates

The invariance of the Lagrangian density un-der a global phase transformation ψ → ψe−iϑ ofthe complex scalar field gives (via the Noether’stheorem) a locally conserved current ∂µj

µ = 0,and a conserved charge (number of particles). Weneed to study the number of particles because itis essential to determine whether the configura-tions are stable or not. A necessary requirementtowards the stability of the configurations is a neg-ative binding energy (BE = M − mN), i.e. the

8

star must be energetically more favorable than agroup of unbound particles of equal mass. Fromthe Noether theorem, the current jµ is given by

jµ =i

2

√

| g | gµν [ψ∗∂νψ − ψ∂νψ∗] . (33)

and the number of particles is

N :=

∫

j0d3x. (34)

For the total gravitational mass of localized solu-tions, we may use Tolman’s expression (Tolman1934), or equivalently, the Schwarzschild mass:

M =

∫

(2T 00 − T µ

µ )√

| g | d3x . (35)

In Fig. 5, we show the mass of a boson starwith Λ = 0 as a function of x. We have fitted thiscurve with a Boltzmann-like function

Mfit(x) = A2 +A1 −A2

1 + e(x−x0)/∆x, (36)

which is reliable except in regions very near thecenter, where Mfit(x) becomes slightly negative.The χ2-parameter of the fitting is around 10−5,and values for A1, A2, and ∆x are given in the fig-ure. It is this formula what we have used in Fig.2 to analytically get the σr(p) dependence of theboson star (in the range we use the approximation,the actual mass and the fitting differ negligibly).Note also what the fitting is physically telling us:it represents a black hole of mass A1 plus an in-ner exponentially decreasing correction. This isthe first time that such a Boltzmann-like fitting isdone, and we think it could be usefully applied inother analytical computations. Models with dif-ferent Λ and σ can be equally well fitted.

Since boson stars are prevented from gravita-tional collapse by the Heisenberg uncertainty prin-ciple, we may make some straightforward massestimates (Mielke and Schunck 1998): For a bo-son to be confined within the star of radius R0,the Compton wavelength has to satisfy λψ =(2π~/mc) ≤ 2R0. In addition, the star radiusmust be of the order of the last stable Kepler or-bit 3RS around a black hole of Schwarzschild ra-dius RS := 2GM . In the case of a mini-bosonstar of effective radius R0

∼= (π/2)2RS close to itsSchwarzschild radius one obtains the estimate

Mcrit∼= (2/π)m2

Pl/m ≥ 0.633M2Pl/m . (37)

The exact value in the second expression wasfound only numerically.

For a mass ofm = 30 GeV, one can estimate thetotal mass of this mini–boson star to be M ≃ 1010

kg and its radius R0 ≃ 10−17 m, amounting adensity 1048 times that of a neutron star. In thecase of a boson star (λ 6= 0), since |ψ| ∼ mPl/

√8π

inside the boson star (Colpi et al. 1986), the energydensity is

ρ ≃ m2m2Pl (1 + Λ/8) . (38)

Equivalently, we may think that this correspondsto a star formed from non–interacting bosons withre-scaled mass m → m/

√

1 + Λ/8 and conse-quently, the maximal mass scales with the cou-pling constant Λ as,

Mcrit ≃2

π

√

1 + Λ/8m2

Pl

m. (39)

There is a large range of values of mass and radiusthat can be covered by a boson star, for differentvalues of Λ and σ0. For instance, if m is of theorder of the proton mass and λ ≃ 1, this is in therange of the Chandrasekhar limiting mass MCh :=M3

Pl/m2 ≃ 1.5M⊙.

Larger than these estimates is the range ofmasses that a non-topological soliton star pro-duces, this is because the power law depen-dence on the Planck mass is even higher: ∼10−2(m4

Pl/mΦ20). These configurations are static

and stable with respect to radial perturbations.It is by no means true, however, that these arethe only stable equilibrium configurations thatone can form with scalar fields. Many extensionsof this formalism can be found. It is even pos-sible to see that boson stars are a useful settingwhere to study gravitation theory in itself (Torres1997; Torres et al. 1998c,d; Whinnett and Tor-res 1999). Most importantly for our hypothesisis that rotating stable relativistic boson stars canalso be found with masses and radii comparable inmagnitude to their static counterparts (Schunckand Mielke 1996; Yoshida and Eriguchi 1997). Inastrophysical settings it is usual to expect someinduced rotation of stellar objects, and it is im-portant that these rotation may not destabilizethe structure. Other interesting generalization isthat of electrically charged boson stars introducedby Jetzer and van der Bij (1989). Although itis usually assumed that selective accretion will

9

quickly discharge any astrophysical object, somerecent results by Punsly (1998) suggest this maynot always be the case.

From these simple considerations, we can seta difference between boson and fermion conden-sations (e.g. neutrino condensation): in the sec-ond case we can have an extended object (in thecase of Sgr A∗, its size can be of about ∼ 30ld)and it can be quite diluted. In the case of a bo-son star, the object is “strongly” relativistic, ex-tremely compact, and its size is comparable withits Schwarzschild radius. We shall discuss the con-sequences of this point widely in the next section.

4.3. Effective potentials

The motion of test particles can be obtainedfrom the Euler-Lagrange equations taking into ac-count the conserved canonical momenta (Shapiroand Teukolsky 1983). In a general spherically sym-metric potential, the invariant magnitude squaredof the four velocity (u2 ≡ gµν x

µxν), which is 1 forparticles with non-zero rest mass and 0 for mass-less particles, yields to

r2 =1

grr

[

E2∞

gtt− u2 − l2

gφφ

]

, (40)

where l and E∞ are constants of motion given by−gφφ sin2 θ φ and gttt (angular momentum and en-ergy at infinity, both per unit mass) respectively,and a dot stands for derivation with respect toan affine parameter. This equation can be trans-formed to

1

2r2 + Veff =

1

2E2

∞e−µ−ν , (41)

where Veff is an effective potential. This namecomes from the fact that in the Schwarzschild so-lution e−µ−ν ≡ 1 and thus, the previous equationcan be understood as a classical trajectory of aparticle of energy E2

∞/2 moving in a central po-tential. This is not so in a more general sphericallysymmetric case, like these non-baryonic stars. Inthe boson star case, for instance, typical met-ric potentials are shown in Fig. 6; note thatfor them e−µ−ν 6= 1 We can see, however, thate−µ−ν < e−µ(0)−ν(0) = C, where C is a constant,and then, usual classical trajectories can be lookedat, in the sense that we may construct an equationof the form r2/2 + Veff < 1/2E2

∞e−µ(0)−ν(0), and

it will be always satisfied.

The effective potential that massive or masslessparticles would feel must be very different from theblack hole case. In the case of a massless particle,the effective potential is given by

Veff = e−µl2

2r2, (42)

while for a massive test particle it is

Veff =1

2e−µ

(

1 +l2

r2

)

. (43)

We show both, boson (for the case of Λ = 0) andblack hole potentials in Figs. 7 and 8. The massof the central object is fixed to be the same inboth cases (see captions) and the curves representa fourth order Runge-Kutta numerical integrationof the equations we previously derived with Λ = 0and σ(0) = 0.1. In the case of massive test parti-cles we use l2/M2 = 0, 12 and 15; thus explicitlyshowing the change in the behavior of Veff for theblack hole case. As l increases, this shape changesfrom a monotonous rising curve to one that hasa maximum and a minimum before reaching itsasymptotic limit. These extrema disappear forl2/M2 < 12. In the case of boson stars, however,if l 6= 0 we have a divergence in the center at r = 0and only one extremum -a minimum-, which oc-curs at rising values of r/M as l grows. The curveVeff for l = 0 -radially moving objects- is not diver-gent, and we can see that the particles may reachthe center of the star with a non-null velocity,given by 1/2r2 = 1/2E2

∞e−µ(0)−ν(0) − Veff(0) > 0,

and will then traverse the star unaffected.

For massless particles, differences are also noto-rious: a black hole produces a negative divergenceand a boson star a positive one. Radial motionof massless particles is insensitive to Veff , beingthis equal to zero, as in the Schwarzschild case.In both cases, for massive and massless particles,outside the boson star the potential mimics theSchwarzschild one.

4.4. Particle orbits

In the case of a black hole, we can use the New-tonian analogy. Orbits can be of three types: ifthe energy is bigger than the effective potential atall points, particles are captured. If the energy issuch that the energy equals the effective potentialjust once, particles describe an unbound orbit, and

10

the point of equality is known as turning point. Ifthere are two such points, orbits are bound aroundthe black hole. Orbits in which the energy equalsthe potential in a minimum of the latter are cir-cular and stable (r = 0, V ′′

eff < 0).

We can not, because of the different relation-ship between the metric potentials we commentedabove, do the same analysis using Veff for the bo-son star. We can, however, note that in most casesthe total equation for the derivative of r will bemodified in a trivial way: If we look at the caseswhere the effective potential has a divergence atthe center (and because the metric coefficients donot diverge), there is no other possibility for theparticles more than found a turning point. Or-bits can then be bound or unbound depending onthe energy, but we can always find a place wherer = 0 and then it has to reverse its sign. In theparticular cases in which the equality happens atthe minimum of the potential we have again sta-ble circular orbits. Only in the case where l = 0particles can traverse the scalar star unaffected.For m 6= 0 there is still the possibility of finding aturning point, if the energy is low enough. How-ever, if the particle is freely falling from infinity,with E∞ = 1, all energy is purely rest mass, it willradially traverse the star, as would do a photon.

We conclude that all orbits are not of the cap-ture type. They can be circular, or unbound, andthey all have at least one turning point. Thishelps to explain why a non-baryonic object willnot develop a singularity while still being a rel-ativistic object (comparable effective potentials,equivalently, comparable relativisticity coefficientGM(r)/r.).

5. Galactic parameters and mass

One interesting fact, which seems to be notreferred before, is the point that for all thesescalar stars, their radius is always related withthe mass in the same way: R & Mm−2

Pl . Thisis indeed the statement -contrary to what it isusually assumed- that not all interesting astro-physical ranges of mass and radius can be mod-eled with scalar fields. In the scalar star models,from the given central mass, the radius we obtainfor the star is comparable to that of the horizon,R ∼ m−2

Pl × 2.61 × 106M⊙ ∼ 3.9 × 1011cm.

The question now is for which values of the pa-

rameters we can obtain a scalar object of such ahuge mass. For the case of mini-boson star weneed an extremely light boson:

m[GeV] = 1.33 × 10−25M(∞)

MBH. (44)

Given a central density σ(0), M(∞) stands for thedimensionless value of the boson star mass as seenby an observer at infinity. MBH is the value ofthe black hole mass (in millions of solar masses),obtained by fitting observational data. Then, weare requiring that the total mass of the boson starequals that of the black hole. For instance, in Fig.1, we have taken Eckart and Genzel’s (1997) andGenzel et al.’s (1996) data, and fit them with amini-boson star with σ(0) = 0.1, which yields to aboson mass given by m[GeV] = 2.81 × 10−26; thetotal mass of the star (without the cluster contri-bution) is 2.5 ×106 M⊙.

In the case of boson stars, and using the criticalmass dependence, ∝

√λm3

Pl/m2, the requirement

of a 2.5 million mass star yields to the followingconstraint,

m[GeV] = 7.9 × 10−4

(

λ

4π

)1/4

. (45)

It is possible to fulfill the previous relationship,for instance, with a more heavy boson of about1 MeV and λ ∼ 1. A plot of this relation -forsome values of λ- is shown in Fig. 9. Note that inthis case, the value of the dimensionless parameterΛ is huge, and special numerical procedures, asexplained above, must be used to obtain solutions.The characteristics of these solutions have provento be totally similar to those with Λ = 0 whichwere used in Fig. 1, just the adimensionalizationdiffers.

Finally, in the case of a non-topological solitonwe obtain the following constraint,

m[GeV] =7.6 × 1012

Φ20[GeV]

2 . (46)

For the usually assumed case, in which the or-der parameter Φ0 is of equal value than the bo-son mass, we need very heavy bosons of unit massm = 1.2×104GeV. Other possible pairs are shownin Fig. 10.

11

5.1. Boson candidates?

Based only on the constraints imposed by themass–radius relationship valid for the scalar starsanalyzed, we may conclude that:

1. if the boson mass is comparable to theexpected Higgs mass (hundreds of GeV),then the Center of Galaxy could be a non-topological soliton star;

2. an intermediate mass boson could producea super-heavy object in the form of a bosonstar;

3. for a mini-boson star to be used as centralobjects for galaxies it is needed the existenceof an ultra-light boson.

These conclusions should be considered as orderof magnitude estimations. Several reasons forceus to make this warning. Firstly, we are just con-sidering static and uncharged stars, this is just amodel (the simplest), but more complicated onescan modify the actual constraints. Secondly, wedo not know the exact form of the self-interaction,or in the case of non-topological stars, the valueof Φ0. For instance, consider the Higgs mass.In the electro-weak theory a Higgs boson doublet(Φ+,Φ0) and its anti-doublet (Φ−, Φ0) are neces-sary ingredients in order to generate masses for theW± and Z0 gauge vector bosons. Calculations oftwo–loop electro-weak effects have lead to an in-direct determination of the Higgs mass (Gambino1998). For a top quark mass of Mt = 173.8 ± 5GeV, the Higgs mass is mh = 104+93

−49 GeV. How-ever, experimental constraints are weak. Fermi-lab’s tevatron (Han and Zhang 1999) has a massrange of 135 < mh < 186 GeV, and together withLHC at Cern, they could decide if these Higgs par-ticles (in the given range) exist in nature. In-teresting is to note that as a free particle, theHiggs boson is unstable with respect to the de-cays h→W+ +W− and h→ Z0 +Z0. However,as remarked by Mielke and Schunck (2000), in acompact object, and in full analogy with neutronstars –where there is equilibrium of β and inverse βdecay– these decay channels are expected to be inequilibrium with the inverse process Z0 +Z0 → h.

We should also mention the possible dilatonsappearing in low energy unified theories, wherethe tensor field gµν of gravity is accompanied

by one or several scalar fields. In string effec-tive super-gravity (Ferrar et al. 1994), for in-stance, the mass of the dilaton can be relatedto the super-symmetry breaking scale msusy bymϕ ≃ 10−3(msusy/ TeV)2 eV. Finally, a scalarwith a long history as a dark matter candidateis the axion, which has an expected light massmσ = 7.4× (107GeV/fσ) eV > 10−11 eV with de-cay constant fσ close to the inverse Planck time.Goldstone bosons have also inferred mass in therange of eV and less, mg < 0.06− 0.3 eV (Umedaet al. 1998).

If boson stars really exist, they could be theremnants of first-order gravitational phase tran-sitions and their mass should be ruled by theepoch when bosons decoupled from the cosmolog-ical background. The Higgs particle, besides itsleading role in inflationary theories, should be thebest and natural candidate as constituent of a bo-son condensation if the phase transition occurredin early epochs. A boson condensation should beconsidered as a sort of topological defect relic. Inthis case, as we have seen, Sgr A∗ could be a soli-ton star. If soft phase-transitions took place dur-ing cosmological evolution (e.g. soft inflationaryevents), the leading particles could have been in-termediate mass bosons and so our supermassiveobjects should be genuine boson stars. If the phasetransitions are very recent, the ultra-light bosonscould belong to the Goldstone sector giving rise tomini-boson stars.

We shall not discuss here any further the for-mation processes of boson stars. The reader is re-ferred to the recent paper by Mielke and Schunck(2000) and references therein. It is apparent thatfor every possible boson mass in the particle spec-trum there is a boson star model able to fit thegalactic center constraints, at least in order ofmagnitude.

6. Accretion and luminosity

6.1. Relativistic rotational velocities

For the static spherically symmetric metric con-sidered here circular orbit geodesics obey,

v2ϕ =

rν′eν

2= eν

eµ − 1

2+

8π

m2Pl

prr2 eµ+ν

2

≃ M(r)

r+

8π

m2Pl

prr2 eµ+ν

2. (47)

12

These curves increase up to a maximum followedby a Keplerian decrease. Liddle and Schunck(1997) found that the possible rotation veloci-ties circulating within the gravitational boson starpotential are quite remarkable: their maximumreaches more than one-third of the velocity of light.Schunck and Torres (2000) proved that these highvelocities are quite independent of the particu-lar form of the self-interaction and are usuallyfound in general models of boson stars. For in-stance, for Λ = 0, 300 of the Colpi et al. ’s(1986) choice, Ucosh = αm2

[

cosh(β√

|ψ|2) − 1]

,

and Uexp = αm2[

exp(β2|ψ|2)−1]

, the maximal ve-locities are: 122990 km/s at x = 20.1 for Λ = 300,102073 km/s at x = 4.1 for Λ = 0, 104685 km/sat x = 4.2 for Uexp, and 102459 km/s at x = 5.9for Ucosh (Schunck and Torres 2000).

With such high velocities, the matter possessesan impressive kinetic energy, of about 6% of therest mass; i.e. to obtain the required luminosity wewould need that about 10−8 to 10−7 solar massesper year be transformed into radiation. Note thatthe required matter-radiation transfer is at leasttwo orders of magnitude smaller than the accretionrate towards Sgr A∗.

The maximum rotational speed is attained welloutside the physical radius of the star, as can beseen by computing the dimensionless x value forthe star radius (e.g. it happens between x ∼ 5and 15 for Λ going from 0 to 300). It is interestingto note also that the dependence of the maximumvelocity on Λ is not very critical, and the sameprocess can be operative with mini-boson stars.The rotational velocity is dependent on the cen-tral density, increasing with a higher value of σ0.To obtain large rotational velocities, it is neededthat the central density of the star be highly rela-tivistic, for Newtonian solutions velocities are lowand quite constant over a larger interval. This isconsistent with the density constraint of the darkobject in Sgr A∗.

6.2. The black hole danger

How can one justify that the accretion onto thecentral object –a neutrino ball or a boson star– willnot create a black hole in its center anyway. Inter-stellar gas and stars, while spiraling down towardsthe center of the object, will begin to collide witheach other, and may glue together at the center,

what could be the seed for a very massive blackhole. Even a black hole of small mass can spiralinwards, and if it remains at the center, that blackhole itself could be the seed. On the other hand,stellar formation of massive stars would yield, af-ter evolution, to a black hole. Then, we need toconsider whether there is a mechanism that pre-vents the formation of a very massive baryonicobject -leading inevitably to a black hole- in thecenter of the galaxy. Below we provide an answerto this question.

The key aspect to consider here is disruption. Astar interacting with a massive object can not betreated as a point mass when it is close enough tothe object such that it becomes vulnerable to tidalforces. Such effects become important when thepericentre rmin is comparable to the tidal radius(Rees 1988),

rt = 5 × 1012M1/36

(

R∗

R⊙

) (

M∗

M⊙

)−1/3

cm. (48)

Here, R∗,⊙ stands for the radius of the star and thesun respectively, while M∗,⊙ for the masses. M6 isthe mass of the central object in millions of solarmasses. rt is the distance from the center objectat which M/r3 equals the mean internal energyof the passing star. Alternatively, we may com-pute the Roche criterion, a comparison betweenthe smoothed out mass of the disruptor and the in-ternal mean energy of the disruptee (Krolik 1999).This catastrophe radius is

Rc ∼ Rg 2

(

M

108M⊙

)−2/3

(ρ⋆/ρ⊙)−1/3, (49)

where Rc is given in units of the event horizon,Rg, of the black hole, and the falling star has den-sity ρ∗. Only for black holes with masses smallerthan 108M⊙ and for certain low density stars -redgiants- we can expect that the disruption happensoutside the event horizon. This is the reason whysupplying the material at lower densities, with theblack hole gravity dominating the situation, cangenerate more power. Rees has also given an es-timate of how frequently a star enters this zone.When star velocities are isotropic, the frequencywith which a solar-like star pass within a distancermin is

∼ 10−4M4/36

(

N∗

105pc−3

) (

σ

100km s−1

) (

rmin

rt

)

yr−1,

(50)

13

where N∗ is the star density and σ the velocitydistribution. Disruptions are rare events that hap-pens once in about 10000 years.

6.2.1. Scalar stars

Because of the similar metric potentials, farfrom the center of the non-baryonic star, the ac-cretion mechanism will be the same than that op-erative in the Schwarzschild case. If a boson staris in the center of the galaxy, the characteristicsof the tidal radius and the timescale of disruptionoccurrence will be similar to those of a black holeof equal mass. Stars falling inwards will all bedisrupted after they approach a minimum radius.Contrary to black holes of big masses, which canswallows stars as a whole and disrupt them be-hind their event horizons, boson stars will disruptall stars –most stars outside and a few inside theSchwarzschild radius of a black hole of equal mass–at everyone sight.

In the case of black holes, the most recent sim-ulations (Ayal et al. 2000) show that up to 75%of the mass that once formed the disrupted starbecome unbound. For boson stars, we argue thatonce the star is disrupted to test particles (withmasses absolutely negligible to that of the centralobject), and because there is no capture orbits,all particles follow unbound trajectories. In thissense, all material is diverted from the center andthe formation of a black hole is avoided.

It could be worth of interest to perform numer-ical simulations changing the central object froma black hole to a boson star, to see the aftermathand the fate of the debris in an actual boson-star-generated disruption. This, however, could wellnot be an easy task: In black hole simulations, theminimum radius is maintained still far from thecenter (∼ 10 Schwarzschild radius), where Newto-nian or Post-Newtonian approximations are valid.To actually see the difference between a black holeand a boson star it could be necessary to attaininner values of radii, where the behavior is com-pletely relativistic.

One case could merit further attention: the pos-sible spiraling of a black hole of stellar size. Thiscase may complicate the situation since a blackhole can not be disrupted. However, differences inmasses are so large that it will behave as a testparticle for the boson central potential, and will

be also diverted from the center. Moreover, be-ing of stellar size, it will be appreciably influencedby other intruding stars, also making it to left anstatic position at the boson star center.

Finally, it is worth of interest to study otherpossible observational consequences of boson stardisruption. The ejecta now is 100% of the starmass, and may possible produce some long termeffects in the surrounding medium. In black holescenarios, the bound debris will create a flare asit accretes. Does the same happen here? Whilewe defer this kind of analysis to another work,we mention that if boson stars exists, disruption,what was once thought of as an inevitable con-comitant of black holes, could now happen in non-baryonic environments.

6.2.2. Neutrino balls

If the center of the galaxy is a neutrino ball,one also has to obtain a mechanism that preventsthe formation of a very massive baryonic object.However, we have to expect very crucial differ-ences with a boson star case, caused by the factthat a neutrino ball is an extended object and thatthe gravitational potential is shallower. The firstthing to note is that rt is well within the neu-trino ball, and then, stars will traverse the exte-rior parts of the ball without being disrupted. Indoing so, however, the central mass that they seeat the center will be less than the total mass of theball, and at a distance r = rt the mass enclosed isnegligible. Disruption can not proceed and othermechanism have to devised. We note then thatthe observation of disruption processes in the cen-ter of a galaxy is then indicative that a neutrinoball is not there.

When the mass enclosed by the neutrino ballis small enough (say, O(103)M⊙), the accretiondisk will be unstable. This happens about 0.1 –1 light years from the center. There, stars whichcould actually form at a rate of 1 per hundredthousand years (the actual number will depend onthe mass of the star) will be probably kicked off byintruding stars (Viollier 2000). The absence of thedisruption mechanism makes this problem worse,since given an enough amount of time, it is hardto think in compelling reasons by which gas andstars are expelled from the center.

In a published paper, Tsiklauri and Viollier

14

(1999) suggested that matter arriving at the cen-ter would be diverted in the form of non-radiatingjets generated by pressure of the inner accretiondisk. However, it is not clear that gravity attrac-tive forces of the spiraling objects will be smallerthan gas pressure exerted by the disk. Moreover,even when the baryonic mass acquired by the cen-tral object during the entire age of the universe isat least two orders of magnitude smaller than themass of the central neutrino ball, one should beworried if this yields to a black hole of that mass,since it may be the seed for a further collapse, orappreciably influence the dynamics. The mecha-nism considered in the previous paragraph seemsmore adequate than this for solving this question,if that is finally possible. To us, it is yet an unclearissue in the neutrino ball scenario.

7. Discussion: how to differentiate among

these models?

One of the easiest things one may think of is tofollow the trajectory of a particular star. This hasbeen done by Munyaneza et al. (Munyaneza et al.2000) in the case of a neutrino ball. The trajec-tory of S1, a fast moving star near Sgr A∗, offersthe possibility of distinguishing between a blackhole and a neutrino condensate, since Newtonianorbits deviate from each other by several degreesin a period of some years. However, as soon as thecentral object is not so extended, as in the bosonstar case, this technique is useless (in every casein which the pericentre is far than the tidal ra-dius), and other forms of detecting their possiblepresence have to be devised.

It has been already noted that X-ray astronomycan probe regions very close to the Schwarzschildradius. It is only observations in the X-ray bandthat can study the inner accretion disk as closeto the center as an event horizon would be. Re-cent results from the Japanese-US ASCA missionhave revealed a broadened iron line feature thatcomes from so close to the event horizon that agravitationally redshift is observed. This is 10000times closer into the black hole than what canbe pictured by HST. In particular, Iwasawa etal. (1996) claimed that ASCA observations toSeyfert 1 galaxy MCG-6-30-15 got data from 1.5gravitational radii, and conclude that the peculiarline profile suggests that the line-emitting region is

very close to a central spinning (Kerr) black holewhere enormous gravitational effects operate. Bythe way, this is stating that a neutrino ball cannot be the center of that galaxy. However, as wasalready noted (Schunck and Liddle 1997), a bosonstar could well be a possible alternative, and X-ray could be used to map out in detail the formof the potential well. The NASA Constellation-X (Constellation-X web page 2000) mission, to belaunched in 2008, is optimized to study the iron Kline feature discovered by ASCA and, if they arethere, will determine the black hole mass and spinfor a large number of systems. Still, Constellation-X will provide an indirect measure of the proper-ties of the region within a few event horizon radii.A definite answer in this sense will probably begiven by NASA-planned MAXIM mission (Maximweb page 2000), a µ-arcsec X-ray imaging mission,that would be able to take direct X-ray picturesof regions of the size of a black hole event horizon.Both of these space mission will have the abilityto give us proofs of black hole existence, or to pro-vide evidence for more strange objects, like bosonstars.

Very recently, Falcke et al. (2000) have notedthat gravitational lensing observations of verylarge baseline interferometry (VLBI) could givethe signature to discriminate among these mod-els. Falcke et al. assumed that the overall specificintensity observed at infinity is an integration ofthe emissivity (taken as independent of the fre-quency or falling as r−2) times the path lengthalong geodesics. Defining the apparent bound-ary of a black hole as the curve on the sky planewhich divides a region where geodesics intersectsthe horizon from a region whose geodesics missthe horizon, they noted that photons on geodesicslocated within the apparent boundary that canstill escape to the observer will experience stronggravitational redshift and a shorter total pathlength, leading to a smaller integrated emissivity.On the contrary, photons just outside the appar-ent boundary could orbit the black hole near thecircular photon radius several times, adding tothe observed intensity. This is what produces amarked deficit of the observed intensity inside theapparent boundary, which they refer to as the“shadow” of the black hole. The apparent bound-ary of the black hole is a circle of radius 27Rgin the Schwarzschild case, which is much larger

15

than the event horizon due to strong bending oflight by the black hole. This size is enough toconsider the imaging of it as a feasible experimentfor the next generation of mm and sub-mm VLBI.While the observation of this shadow would con-firm the presence of a single relativistic object, anon-detection would be a major problem for thecurrent black hole paradigm.

Concerning the ideas put forward in this work,Falcke et al.’s shadow concept is appealing. Inthe case of a boson star, we might expect somediminishing of the intensity right in the center,this would be provided by the effect on relativisticorbits, however, this will not be as pronouncedas if a black hole is present: for that case, manyphotons are really gone through the horizon andthis deficit also shows up in the middle. If a bosonstar is there, some photons will traverse it radially,and the center region will not be as dark as in theblack hole case. A careful analysis of Falcke et al.’sshadow behavior replacing the central black holewith a boson star model would be necessary to getany further detail, and eventually an observableprediction.

We also mention that the project ARISE (Ad-vanced Radio Interferometry between Space andEarth) is going to use the technique of Space VLBIto increase our understanding of black holes andtheir environments. The mission, to be launchedin 2008, will be based on a 25-meter inflatablespace radio telescope working between 8 and 86GHz (Ulvestad 1999). It will study gravitationallenses at resolutions of tens of µarcsecs, yieldinginformation on the possible existence (and signa-tures) of compact objects with masses between103M⊙ and 106M⊙.

Another possible technique for detecting bo-son stars from other relativistic objects could begravitational wave measurements (Ryan 1997). Ifa particle with stellar mass is observed to spi-ral into a spinning object with a much largermass and a radius comparable to its Schwarzschildlength, from the emitted gravitational waves, onecould in principle obtain the lowest multipole mo-ments. The black hole no-hair theorem says thatall moments are determined by its lowest two,the mass and angular momentum (assuming thecharge equal to zero). Should this not be so,the central object would not be a black hole,and as far as we know, the only remaining vi-

able candidate would be a boson star. In tenyears time, perhaps, a combination of gravita-tional wave measurements, better determinationof stellar motions, and mm and sub-mm VLBItechniques could give us a definite picture of thesingle object at the center of our Milky Way.

From a theoretical point of view, developmentson gravitational lensing theory in very strong fieldregimes will be of extreme importance since, forobjects like Sgr A∗, the standard weak field theorydoes not hold and new effects have to be expected(Virbhadra and Ellis 2000).

8. Conclusions

We have shown that boson stars provide the ba-sic necessary ingredients to fit dynamical data andobserved luminosity of the center of the Galaxy.They constitute viable alternative candidates forthe central supermassive object, producing a the-oretical curve for the projected stellar velocitydispersion consistent with Keplerian motion, rel-ativistic rotational velocities, and having an ex-tremely small size.

Other singularity-free models were as well con-sidered, as non-baryonic fermion stars (e.g. neu-trino, or gravitino condensations). In this case,the object is sustained by its Fermi energy, whilein the boson star case, it is the Heisenberg’s Un-certainty Principle which prevents the system fromcollapsing to a singularity. Due to this fact, bo-son stars are genuine relativistic objects where astrong gravitational field regime holds.

This difference in the relativistic status of bothobjects is not trivial. While fermion neutrino ballsare extended objects, boson stars mimic a blackhole. Disruption processes can not happen in afermion condensation, and it has the unpleasantconsequence of not providing with a straightfor-ward mechanism by which stars could be divertedfrom the center, and through which finally avoidthe formation of a massive black hole inside thecondensate.

The formation of boson stars, neutrino balls,and black holes can all be competitive processes.Then, it might well be that even if we discover thata black hole is in the center of the galaxy, othersgalaxies could harbor non-baryonic centers. In thecase of boson stars, only after the discovery of theboson mass spectrum we shall be in position to de-

16

termine a priori which galaxies could be modeledby such a center. Observations of galactic centerscould then suggest the existence of boson scalarsmuch before than their discovery in particle physi-cists labs.

We acknowledge insightful comments by H. Fal-cke, F. Schunck, D. Tsiklauri, R. Viollier, and A.Whinnett. Also, we acknowledge informal talkswith D. Bennet, R. di Stefano, C. Kochanek, andA. Zaharov at the Moriond 2000 meeting. D.F.T.was supported by CONICET as well as by fundsgranted by Fundacion Antorchas, and particularlythanks G. E. Romero for his encouragement andcriticism. S. C. and G. L.’s research was sup-ported by MURST fund (40%) and art. 65 D.P.R.382/80 (60%). G.L. further thanks UE (P.O.M.1994/1999).

17

A. Appendix

For reader’s convenience, we quote here the dimensional conversion for the radius and the mass of a bosonstar. Using the value of 1 GeV in cm−1, and taking into account the dimensionless parameter x = mr, weget

r[pc] =x

m[GeV]6.38 × 10−33. (A1)

For the mass, recalling that M = M(x)m2Pl/m, we get

M [106M⊙] =M(x)

m[GeV]1.33 × 10−25. (A2)

In the case where Λ ≫ 1, both right hand sides of the previous formulae get multiplied by Λ1/2.

18

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Fig. 1.— Enclosed mass in the center of the galaxytogether with observational data points. See dis-cussion in the main text.

Fig. 2.— Projected velocity dispersions: observa-tional data and fit using a boson star model. Seediscussion in the main text.

Fig. 3.— Mass and number of particles, in dimen-sionless units, for a Λ = 10 boson star (Colpi etal. (1986)) potential.

Fig. 4.— Stability analysis for Λ = 10 configura-tions. Catastrophe theory ensures us that the firstbranch, which includes the (0,0) point, is the onlystable one.

Fig. 5.— Boson star mass as a function of x, for aΛ = 0, σ(0) = 0.1 model. We show a Boltzmann-like fitting (dotted curve, but almost everywhereon the solid curve) and its residue (solid lowerline).

Fig. 6.— Boson star metric potentials grr = eµ

(dashed) and gtt = eν (solid). Boson star param-eters are Λ = 0, σ(0) = 0.1.

Fig. 7.— Boson (solid) and Schwarzschild(dashed) effective potentials for massless particles,b = l2/E2

∞. The mass for both, the black hole andthe boson star, was taken as M = 0.62089m2

Pl/m.The maximum in the black hole case happens, in-dependently of l, for r/M = 3.

Fig. 8.— Boson (solid) and Schwarzschild(dashed) effective potentials in the case of mas-sive particles. The mass of the central object isas in the previous figure. The three curves corre-spond to l2/M2 = 0, 12 and 15 (from bottom totop).

Fig. 9.— Constraint in the boson star fundamen-tal parameters which gives rise to an object of twomillion solar masses within approximately ten so-lar radius, consistent with the mass of the centralobject in our galaxy.

Fig. 10.— Constraint in the non-topological soli-ton fundamental parameters which gives rise to anobject consistent with the mass of the central ob-ject in our galaxy. It is specially marked the usualcase in which Φ = m.

21

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